© 2013 pearson education, inc.. let’s just start with the dot product formula 12j the scalar...


Let’s just start with the Dot Product formula

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The scalar product is a manner of multiplying vectors.

The scalar product of two vectors can be constructed by:

taking the component of one vector

in the direction of the other and

multiplying it times the magnitude of the other vector.


The Definition of the Dot Product of Two Vectors

The dot product of u = and v = is

u1,u2

v1,v2

uv u1v1 u2v2Ex.’s Find each dot product.

a. 4,5 2,3

b. 2, 1 1,2

c. 0,3 4, 2

4 2 5 3 23

2 1 1 2 0

0 4 3 2 6

Let’s just start with the Dot Product formula

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Basically, the scalar product (A.K.A. the Inner Product)

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1 1

2 2

3 3

and w

v w

v v w

v w

1 1 2 2 3 3v w v w v w v w

Read as, “V dot W”


If you want to know more about the dot product

check out the videos on the IB class website

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What does this mean?

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• Normal multiplication combines growth rates

• “3 x 4″ can mean “Take your 3x growth and make it 4x larger (i.e., 12x)”.

• A vector is “growth in a direction”.

• The dot product lets us apply the directional growth of one vector to another

• The result is how much we went along the original path (positive progress, negative, or zero).



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Seeing Numbers as vectorsLet’s start simple, and see 3 x 4 as a dot product:

• 3 is “directional growth” in a single dimension (x-axis, let’s say)

• 4 is “directional growth” in that same direction.

• 3 x 4 = 12 means 12x growth in that single dimension. Ok?



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• Suppose each number refers to a different dimension.

• 3 means “triple your bananas” (sigh… or “x-axis”)

• 4 means “quadruple your oranges” (y-axis).

• They’re not the same type of number: what happens when we apply growth, aka use the dot product, in our “bananas, oranges” universe?

”



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Seeing Numbers as vectorsLet’s start simple, and see 3 x 4 as a dot product:• (3,0) is “Triple your bananas, destroy oranges”

• (0,4) is “Destroy your bananas, quadruple oranges



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Applying (0,4) to (3,0) means

•“Destroy banana growth, quadruple orange growth”.

•But (3, 0) had no orange growth to begin with

•The net result is 0 (“Destroy all your fruit, buddy”).



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See how we’re “applying” and not adding?

With addition, we sort of smush the items together: (3,0) + (0, 4) = (3, 4) [a vector which triples your oranges and quadruples your bananas].

“Application” is different. We’re mutating the original vector according to the rules in the second. And the rules are “Destroy your banana growth rate, and triple your orange growth rate“. And, sadly, this leaves us with nothing.


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Mario-Kart Speed BoostIn Mario Kart, there are “boost pads” on the ground that increase your speed (Never played? I’m sorry.)


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Imagine the red vector is your speed (x & y direction), and the blue vector is the orientation of the boost pad (x & y direction). Larger numbers are more power.How much boost will you get? For the analogy, imagine the pad multiplies your speed:


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If you come in going 0, you’ll get nothing [if you are just dropped onto the pad, there’s no boost]If you cross the pad perpendicularly, you’ll get 0 [just like the banana obliteration, it will give you 0x boost in the perpendicular direction]


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But, if we have some overlap, our x-speed will get an x-boost, and our y-speed gets a y-boost:


Better?

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So, you’re sitting there and you ask yourself, “Self, how do I find the angle between vectors?”

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Great Question!


Consider the following:

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v

1

2

3

v

v v

v

w

1

2

3

w

w

w

w

Translate one of the vectorsso that they both start at the same point



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This is the vector:

-v + w = w - v

v

w

θ

Has length:

|w – v|



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v

w

θ |w – v|

From the Law of Cosines where c is the side opposite the angle theta:

2 2 2 2 cosc a b ab

2 2 2| | | | | | 2 | || | cosw v v w v w


However,

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v

w

θ|w – v|

2 2 2| | | | | | 2 | || | cosw v v w v w

1 1 1 1

2 2 2 2

3 3 3 3

- =

w v w v

w v w v w v

w v w v


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1 1 1 1

2 2 2 2

3 3 3 3

- =

w v w v

w v w v w v

w v w v

2 2 2

1 1 2 2 3 3 = w v w v w v

2 2 2| | | | | | 2 | || | cosw v v w v w

2 2 2 2 2 2

1 2 3 1 2 3 + 2 | || || cosv v v w w w v w


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2 2 2| | | | | | 2 | || | cosw v v w v w

1 1 2 2 3 3 | || || cosv w v w v w v w

2 2 2 2 2 2

1 2 3 1 2 3 + 2 | || || cosv v v w w w v w

| || || cosv w v w


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cos| || |

v w

v w

So, to find the angle between vectors can be found using:


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v w w v

Algebraic Properties of the Scalar Product

2| |v v v

( )v w x v w v x

( ) ( )v w x y v x v y w x w y


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0v w Other Properties of the Scalar Product

If v and w are perpendicular or “orthogonal”

| | | || |v w v w

If v and w are non-zero parallel vectors


HOMEWORK

Test Next FridayOver chapter 12 and

Two-ish problems from chapter 6


HOMEWORK

page 310(12j)Numbers 1 – 23

Skip #9

Review 12A and 12BOn 12B (Skip #12)

© 2013 pearson education, inc.. let’s just start with the dot product formula 12j the scalar...

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